Derived counterparts of fusion categories of quantum groups

In this text, we study derived versions of the fusion category associated to Lusztig's quantum group $\textbf{U}_q$. The categories that so arise are non-semisimple but recovers the usual fusion ring when passing to complexified Grothendieck rings. On the derived level it turns out that it is possible to define fusion for $\textbf{U}_q$ without using the notion of tilting modules. Hence, we arrive at a definition of the fusion ring that makes sense in any spherical category. We apply this new definition to the small quantum group and we related with some rings of A. Lachowska

Higher Auslander-Reiten sequences and tt-structures

Let $R$ be an artin algebra and $\mathcal{C}$ an additive subcategory of $\operatorname{mod}(R)$. We construct a $t$-structure on the homotopy category $\operatorname{K}^{-}(\mathcal{C})$ whose heart $\mathcal{H}_{\mathcal{C}}$ is a natural domain for higher Auslander-Reiten (AR) theory. The abelian categories $\mathcal{H}_{\operatorname{mod}(R)}$ (which is the natural domain for classical AR theory) and $\mathcal{H}_{\mathcal{C}}$ interact via various functors. If $\mathcal{C}$ is functorially finite then $\mathcal{H}_{\mathcal{C}}$ is a quotient category of $\mathcal{H}_{\operatorname{mod}(R)}$. We illustrate the theory with two examples: Iyama developed a higher AR theory when $\mathcal{C}$ is a maximal $n$-orthogonal subcategory, see \cite{I}. In this case we show that the simple objects of $\mathcal{H}_{\mathcal{C}}$ correspond to Iyama's higher AR sequences and derive his higher AR duality from the existence of a Serre functor on the derived category $\operatorname{D}^b(\mathcal{H}_{\mathcal{C}})$. The category $\mathcal{O}$ of a complex semi-simple Lie algebra $\mathfrak{g}$ fits into higher AR theory by considering $R$ to be the coinvariant algebra of the Weyl group of $\mathfrak{g}$.Comment: 26 pages, accepted for publication in Journal of Algebra 201

The n-th prime asymptotically

A new derivation of the classic asymptotic expansion of the n-th prime is presented. A fast algorithm for the computation of its terms is also given, which will be an improvement of that by Salvy (1994). Realistic bounds for the error with \li^{-1}(n), after having retained the first m terms, for $1\le m\le 11$, are given. Finally, assuming the Riemann Hypothesis, we give estimations of the best possible $r_3$ such that, for $n\ge r_3$, we have $p_n> s_3(n)$ where $s_3(n)$ is the sum of the first four terms of the asymptotic expansion

Tuples of polynomials over finite fields with pairwise coprimality conditions

Let q be a prime power. We estimate the number of tuples of degree bounded monic polynomials (Q1, . . . , Qv) ∈ (Fq[z])v that satisfy given pairwise coprimality conditions. We show how this generalises from monic polynomials in finite fields to Dedekind domains with a finite norm

Higher Education Decisions in Peru: On the Role of Financial Constraints, Skills, and Family Background

"This paper analyzes the relative importance of short term financial constraints vis a vis skills and other background factors affecting schooling decisions when explaining access to higher education in Peru. We focus on college access disparities between rich and poor households. We use a novel household survey that includes special tests to measure cognitive and non-cognitive skills of the urban population age 14-50. These are complemented with retrospective data on basic education and family socioeconomic conditions in a multinomial model. We find that strong correlation between college enrollment and family income in urban Peru is not only driven by credit constraints, but also by poor college readiness in terms of cognitive skills and by poor family and educational backgrounds affecting preferences for schooling. Family income explains, at most, half of the college access gap between poor and non-poor households. The other half is related to differences in parental education, educational background and cognitive skills. Our results indicate that credit and/or scholarship schemes alone will not suffice to change the regressive nature of higher education enrollment in Peru, and that such programs will face strong equity-efficiency trade-offs."Higher education, Cognitive skills, Non-cognitive skills, Credit constraints, Peru